Tuesday, 29 January 2013

The UK government has recently announced the route for the
second leg of the long awaited high speed rail link between London and the
north of England. This prompted the usual appeals to cost-benefit analysis to
argue one way or the other. Most ‘experts’ seemed to be arguing that the
supposed benefits are largely illusory and so the link does not make economic
sense. But what does that really mean?

Cost-benefit analysis is one of the most basic tools in the
economist’s armoury. It’s pretty clear, however, that the general public don’t
like it. Many people I heard on the radio talking about the rail link were
annoyed by the experts cost-benefit calculations. Similarly, when health
economists argue that a particular cancer drug should not be made available on
the National Health Service because the costs exceed the benefits, people aren’t
too pleased. So, what’s wrong with cost-benefit analysis?

Given that I’m an economist it will be no surprise to hear
that I don’t think anything is wrong with cost-benefit analysis. It’s one of
the most basic principles of economics for a good reason –something is worth
doing if and only the benefit exceeds the cost. What I have much less sympathy for
is the way economists often use cost-benefit analysis.

Doing a proper cost-benefit analysis is difficult because
measuring potential benefits and costs can be a very tricky thing indeed. I’m excited
by a high speed rail link because I want to see technological progress and
development; how can you measure this benefit? Similarly, a cancer drug might
add two years to a person’s life; how can you measure the benefit of that? I
know there are ways we can attempt to measure these things, such as, willingness
to pay and quality adjusted life years, but I also know how imperfect these
measures are!

So, my two concerns with the way economists often do
cost-benefit analysis are as follows: (i) They make life easy for themselves by
counting the simple to measure financial costs and benefits while ignoring the
more difficult to measure costs and benefits; this often biases against the benefits.
(ii) They underplay the huge margin for error in the calculation of costs and
benefits; this leads to a sense of decisiveness that is not justified.

It’s important, therefore, to appreciate the limitations of
cost-benefit analysis. It is a useful tool but it cannot be expected to give clear
cut answers. On tricky issues it is best used as a tool to inform and open
debate rather than as a means to decide. Decisions should come down to questions
of individual or democratic judgement. I, for one, would vote for the high
speed rail link. Maybe you disagree. Just don't think cost-benefit analysis holds all the answers.

Saturday, 19 January 2013

The
issue of standing up at football grounds has been attracting some attention
recently. The basic problem people are talking about is one of externalities
and property rights: Post the Hillsborough disaster, standing is banned at the
major football grounds in Britain. Clearly, however, that does not stop some
fans wanting to stand up in order to enjoy the game that bit more. But, if a
fan stands up that creates a negative externality for anyone sitting behind –
at best the person behind has to also stand up to see the game, at worst the
person cannot see even if they stand up. Some fans are arguing that football is
not football without standing – others are annoyed at not being able to see the
game. My perspective would be to look who has the property rights. And given
that standing at grounds is banned the property rights clearly stand with those
who want to sit and see the game. So, ‘sit down’.

Thoughts of standing at football remind me of watching
football in my childhood years – going around the country to watch Aston Villa.
When the football was not good – you never have long to wait with the Villa – I
would often watch the crowd. And I find crowds fascinating. Indeed, it doesn’t
seem hyperbole to say that I became a game theorist watching football crowds. For
example, one thing you can learn about watching a football crowd are
information cascades. Information cascades are traditionally applied to analyze
consumer choice and the stock market, but a football crowd is just as
interesting. Let me try and explain.

Picture a packed stadium with all the supporters sitting
down comfortably. Then the home team starts attacking and it looks as though
something exciting might happen. If something exciting does happen then the
supporters would rather be standing up in order to let off energy. If nothing
exciting happens they would rather have stayed sitting. In real time, as the
team attacks, each supporter must decide whether to stand up or remain seated.
This scenario has the two key ingredients we need for an information cascade to
occur:

(a) Each supporter has their own beliefs about whether
something exciting might happen. Some may be optimistic, some pessimistic, some
may have a better view, others a worse view, etc. In game theory parlance each
supporter has a private signal of whether something exciting may happen. The
key word here is ‘private’ – only the supporter knows what his signal and
beliefs are.

(b) Choices are made sequentially with the possibility to
observe what others a doing. If a supporter stands up then all the supporters
behind can clearly see that he has stood up. Note, however, that only the
action is observable. The reasons behind the action, i.e. the private signal or
beliefs remain private.

Let’s roll forward time a little until a first supporter
decides to stand up. Suppose his name is Darius. What does Darius’ action – him
standing up – tell us about his signal? Probably a lot. It might be that he is
pessimistic anything exciting will happen and just stood up to go and get a cup
of coffee. Much more likely, however, is that he stood up because he is really
confident something exciting will happen. Suppose that once Darius has stood up
there is a cascade of other people standing up. For example, imagine that Sam,
whose sitting a few rows behind Darius, stands up. What does Sam’s action tell
us about his signal? Probably very little. It could be that Sam is standing up
because he was always confident something exciting might happen. Equally,
however, Sam could be standing up because the actions of Darius and others have
caused him to update his beliefs – initially he was pessimistic something
exciting would happen but has changed his mind. Once we have reached the point
where a Sam’s action tell you nothing about his private signal then we have an
information cascade.

Information cascades have lots of interesting properties.
For example, they mean that mass action can convey very little information.
That Darius stands up tells us something. That the 2,000 supporters around him
stand up tells us very little. The main consequence of this is that information
cascades can be very misleading. If we combined the private signal of every
supporter we might get a good prediction of the chances of something exciting
happening. But, that’s not how it works. Darius triggered the whole thing and
that’s just one private signal which could easily have been wrong. So,
supporters can expect to be up and down like yo-yo’s. Another interesting
property is fragility. For example, suppose that while Darius and the other
2,000 supporters are standing up, Brian stays firmly in his seat. What does
that tell us about Brian’s signal? Potentially quite a lot. It suggests he has
a strong signal that nothing exciting is going to happen – he might, for
instance, have seen that the linesman has flagged to stop play. Given that we
know little about the signal of the 2,000 supporters who stood up we are just
left with a good idea of Darius and Brian’s signals which ‘cancel each other out’.
Brian staying seated can easily, therefore, stop the cascade.

Well that’s the theory. What about the practice. We know
from the fascinating work of Georg Weizsäcker*
that people are biased in situations where information cascades may occur. They
tend to underestimate the information conveyed by Darius or Brian’s action
while overestimating the information conveyed by the 2,000 others who stand up.
Such bias is probably not too surprising. What’s interesting is to know how big
the bias is – how easily are people misled by mass action. My experience of
watching football crowds suggests we are not too easily misled. Indeed, my
impression is that in real situations people have a fairly good intuition of
how information cascades work. For example, my anecdotal evidence, is that you
often get supporters playing the part of Brian by remaining seated while
everyone in front stands, and others reacting to that. This means we get the
fragility that is predicted in theory but unlikely if people are strongly
misled by mass action.

The problem we have is that our theoretical understanding of
information cascades remains largely foccussed on nice textbook cases that are far
removed from real world settings. For example, we know very little about what should
happen in the real time setting that we find in a football ground, or the stock
exchange. This is one area, therefore, of game theory that needs a lot more
work before we can be too confident what is going on. So, football crowds can
teach us something.

Friday, 4 January 2013

Over Christmas I had chance to read The
Stag Hunt and the Evolution of Social Structure by Brian Skyrms. A nice read,
very interesting and thought provoking. There’s a couple of things in the book
that prompt further discussion. The one I want to focus on in this post is the
distinction between the stag hunt game and the prisoners dilemma game.

To be sure what we are talking about, here
is a specific version of both type of game. Adam and Eve independently need to
decide whether to cooperate or defect. The payoff matrix details their payoff
for any combination of choices, where the first number is the payoff of Adam
and the second number the payoff of Eve. For example, in the Prisoners Dilemma,
if Adam cooperates and Eve defects then Adam gets 65 and Eve gets 165.

Prisoners Dilemma

Eve

Cooperate

Defect

Adam

Cooperate

140, 140

65, 165

Defect

165, 65

90, 90

Stag Hunt

Eve

Cooperate

Defect

Adam

Cooperate

140, 140

10, 70

Defect

70, 10

70, 70

The key thing about the prisoners dilemma
is that cooperating is a dominated strategy. It doesn’t matter what Eve does,
it is in Adam’s interest to defect. Similarly, it doesn’t matter what Adam
does, it is in Eve’s interest to defect. So, we have a clear game theoretic
prediction that both Adam and Eve should defect. Simple enough. This result,
however, is a bit depressing given that both Adam and Eve would get much higher
payoffs if they were to cooperate. It’s this trade-off between individual
rationality and collective rationality that has resulted in the prisoners
dilemma, despite its seeming simplicity, being easily the most analyzed game in
game theory. The key questions asked are: (i) whether people cooperate in the
prisoners dilemma, (ii) if they do (many do) then why, and (iii) if they do not
(many do not) then how can we get them to cooperate.

The main thing I liked about Skyrms’ book
is his suggestion that we should focus a little less on the prisoners dilemma
and a little more on the stag hunt game. There are, at least, two reasons to
focus more on the stag hunt game. The reason emphasized by Skyrms is that this
game is often a better description of the applied context we’re interested in than
the prisoners dilemma. A more subtle reason, not explicitly mentioned by Skyrms
but a theme throughout the book nonetheless, is that an understanding of the
stag hunt game can possibly tell us more about the prisoners dilemma than an analysis
of the prisoners dilemma can do. So, what’s different about the stag hunt game?

In this game cooperate is not a dominated
strategy. If Eve cooperates then it is in Adam’s interest to also cooperate.
Which suggests that it should be a lot easier to get cooperation? That,
however, is where things get interesting. If you ask people to play the stag
hunt game then the outcome is remarkably similar to what you get if you ask
people to play the prisoners dilemma. This is the case in the two player
versions given above, or in the more general many player versions (which
correspond to a linear public good game and minimum effort game) where
defection quickly becomes the norm. This empirical finding potentially tells us
a lot. The standard story is that people defect in the prisoners dilemma
because that is the rational thing to do. That story, however, sounds a little
suspect if people defect to a similar extent in the stag hunt game. In the stag
hunt game defection cannot be explained as the ‘rational thing to do’ and is
almost certainly a consequence of people avoiding a risky option. Something
similar may be going on in the prisoners dilemma. If so, it would be a mistake
to put a lack of cooperation in the prisoners dilemma down to defection being
the rational thing to do.

I’m
not saying that different things may not be happening in the prisoners dilemma
and the stag hunt game. Clearly, the problems of obtaining cooperation in the
prisoners dilemma appear greater than in the stag hunt game. My point is more of
a ‘let’s walk before we can run’ nature. It seems ambitious to try and get
people to cooperate in the prisoners dilemma when we don’t know how to get them
to cooperate in the stag hunt game (and we don’t). My hope would be that ways
of obtaining cooperation in the stag hunt game would work pretty well for the
prisoners dilemma as well. And to get cooperation in the stag hunt game the
emphasis must surely be on making people more confident that the person they
are playing with will cooperate. This line of reasoning is quite different to
that found in most of the research on the prisoners dilemma. But, it still
leaves open the question of how to get cooperation in the stag hunt game.
Skyrms had a lot to say on that question, which gives me a nice topic for a
future post.